Subgroups — #27300468

Order 13568.

286 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274887	C₂	1	1	yes	maximal, characteristic
2	#20274879	C₂	1	1	yes	maximal, characteristic
3	#20272814	C₂	1	4	yes	maximal
4	C₄×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133153	—	2	1	no	—
6	C₂×#13133153	C₂²	1	2	yes	—
7	C₂×#13133151	—	2	1	no	—
8	C₂×#13133149	C₂²	1	2	yes	—
9	#16810647	—	2	1	no	—
10	#16810489	—	2	1	no	—
11	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#346	C₄	1	2	yes	complemented
14	C₄×C₁₆×C₅₃	C₄	1	2	yes	complemented
15	C₂×#11719917	—	2	2	no	—
16	C₂×C₄⋊Dic₅₃	—	2	2	no	—
17	#13133153	—	2	4	no	—
18	#13133153	—	4	2	no	—
19	#13133151	—	4	2	no	—
20	#13133149	—	2	4	no	—
21	C₄⋊#88322	—	4	2	no	—
22	#13133124	—	4	2	no	—
23	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
25	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
26	M₃₂×C₅₃	—	2	2	no	—
27	M₃₂×C₅₃	—	2	2	no	—
28	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
29	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
32	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
34	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
35	C₂²×Dic₅₃	—	4	2	no	—
36	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₅₃	D₈	1	2	yes	—
39	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
40	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₈×C₅₃	#41	1	2	yes	—
43	C₂×C₈×C₅₃	#41	1	2	yes	—
44	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
45	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
46	C₄²×C₅₃	SD₁₆	1	2	yes	—
47	C₄⋊Dic₅₃	—	4	2	no	—
48	C₄⋊Dic₅₃	—	4	2	no	—
49	#11719917	—	4	2	no	—
50	#11719917	—	4	2	no	—
51	C₂×#88322	—	4	2	no	—
52	C₁₆×C₅₃	—	2	4	no	—
53	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
54	C₂×C₄×C₅₃	—	4	2	no	—
55	C₂×C₄×C₅₃	#106	1	2	yes	—
56	C₂×C₄×C₅₃	#106	1	2	yes	—
57	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
59	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
60	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
63	C₂×Dic₅₃	—	8	4	no	—
64	C₂×Dic₅₃	—	8	2	no	—
65	C₈×C₅₃	#107	1	2	yes	—
66	C₈×C₅₃	#107	1	2	yes	—
67	C₈×C₅₃	#117	1	2	yes	—
68	C₈×C₅₃	#125	1	2	yes	—
69	#88322	—	8	2	no	—
70	#12887	—	53	1	no	maximal, Sylow
71	C₂²×C₅₃	—	2	2	no	—
72	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
73	C₂²×C₅₃	#486	1	1	yes	characteristic
74	C₂²×C₅₃	#487	1	1	yes	characteristic
75	C₄×C₅₃	—	2	2	no	—
76	C₄×C₅₃	—	2	2	no	—
77	C₄×C₅₃	#382	1	1	yes	characteristic
78	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
79	C₄×C₅₃	#486	1	1	yes	characteristic
80	C₄×C₅₃	#487	1	1	yes	characteristic
81	Dic₅₃	—	16	2	no	—
82	C₄×M₃₂	D₅₃	1	1	yes	characteristic
83	#1848	—	53	1	no	—
84	#1846	—	53	1	no	—
85	#1383	—	53	4	no	—
86	C₂×C₅₃	—	2	1	no	—
87	C₂×C₅₃	—	2	1	no	—
88	C₂×C₅₃	#1834	1	1	yes	characteristic
89	C₂×C₅₃	#2155	1	2	yes	—
90	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
91	C₂×#108	—	53	2	no	—
92	C₂×#108	—	106	1	no	—
93	C₂×#107	—	106	1	no	—
94	C₂×#106	—	53	2	no	—
95	#423	—	106	1	no	—
96	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
97	#382	—	106	1	no	—
98	#346	Dic₅₃	1	2	yes	complemented
99	C₄×C₁₆	Dic₅₃	1	2	yes	complemented
100	C₅₃	#12887	1	1	yes	characteristic, complemented, Sylow
101	C₂×M₁₆	—	106	2	no	—
102	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
103	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
104	C₂×C₄⋊C₄	—	106	2	no	—
105	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
106	M₃₂	—	2	2	no	—
107	M₃₂	—	2	2	no	—
108	C₂×C₁₆	C₂×Dic₅₃	1	4	yes	—
109	#108	—	106	4	no	—
110	#108	—	212	2	no	—
111	#107	—	212	2	no	—
112	#106	—	106	4	no	—
113	C₄⋊C₈	—	212	2	no	—
114	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
115	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
116	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
117	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
118	#95	—	212	2	no	—
119	C₂²×C₄	—	212	2	no	—
120	C₂²×C₄	#11719913	1	2	yes	—
121	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
122	M₁₆	—	212	2	no	—
123	M₁₆	—	212	2	no	—
124	C₂×C₈	—	212	2	no	—
125	C₂×C₈	D₄₂₄	1	2	yes	—
126	C₂×C₈	Dic₂₁₂	1	2	yes	—
127	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
128	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
129	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
130	C₂×C₈	#11719945	1	2	yes	—
131	C₂×C₈	#11719945	1	2	yes	—
132	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
133	C₄⋊C₄	—	212	2	no	—
134	C₄⋊C₄	—	212	2	no	—
135	C₄²	#11719913	1	2	yes	—
136	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
137	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
138	C₁₆	—	2	4	no	—
139	C₂³	C₂×#11719913	1	1	yes	characteristic
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	424	4	no	—
142	C₂×C₄	—	424	2	no	—
143	C₂×C₄	#13133149	1	2	yes	—
144	C₂×C₄	#13133149	1	2	yes	—
145	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
146	C₂×C₄	#13133205	1	1	yes	characteristic
147	C₂×C₄	#13133205	1	1	yes	characteristic
148	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
149	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
150	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
151	C₈	—	424	2	no	—
152	C₈	#13133151	1	2	yes	—
153	C₈	#13133151	1	2	yes	—
154	C₈	#13133176	1	2	yes	—
155	C₈	#13133209	1	2	yes	—
156	C₂²	—	2	2	no	—
157	C₂²	C₂×#13133149	1	1	yes	characteristic
158	C₂²	#16810970	1	1	yes	characteristic
159	C₂²	#16810974	1	1	yes	characteristic
160	C₄	—	2	2	no	—
161	C₄	—	2	2	no	—
162	C₄	—	848	2	no	—
163	C₄	#16810489	1	1	yes	characteristic
164	C₄	C₂×#13133151	1	1	yes	characteristic
165	C₄	#16810970	1	1	yes	characteristic
166	C₄	#16810974	1	1	yes	characteristic
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	#20274843	1	1	yes	characteristic
170	C₂	#20276488	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#13133149
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₄×M₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂×#106
prime radical	C₂³×C₅₃	C₂×SD₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₄×M₃₂	D₅₃
hypocenter	C₅₃	#12887
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
13. C₅₃×#346	C₄	semidirect
14. C₄×C₁₆×C₅₃	C₄	semidirect
98. #346	Dic₅₃	semidirect
99. C₄×C₁₆	Dic₅₃	semidirect
100. C₅₃	#12887	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20274887	2
#20274879	2
#20272814	2
C₄×M₃₂×C₅₃	2
#12887	53